1. **Problem Statement:** A voter can choose among candidates for three positions: - President: Ahmed or Radwa (2 choices) - Vice-President: Karam, Menna, or Eman (3 choices) - Secretary: Fayez, Gana, or Nadia (3 choices) We need to analyze the number of ways to complete the ballot under different conditions. 2. **Tree Diagram Explanation:** The tree has three levels: - Level 1 (President): 2 branches (Ahmed, Radwa) - Level 2 (Vice-President): from each President choice, 3 branches (Karam, Menna, Eman) - Level 3 (Secretary): from each Vice-President choice, 3 branches (Fayez, Gana, Nadia) Total pathways = $2 \times 3 \times 3 = 18$ 3. **Part a: Draw a tree diagram** The tree diagram visually represents all possible voting combinations as described above. 4. **Part b: Number of ways if voter may abstain in some or all elections** If abstaining (not voting) is allowed for each position, add 1 choice ("no vote") per position: - President: $2 + 1 = 3$ - Vice-President: $3 + 1 = 4$ - Secretary: $3 + 1 = 4$ Total ways = $3 \times 4 \times 4 = 48$ 5. **Part c: Number of ways if voter chooses Radwa for President** Fix President choice to Radwa (1 way), Vice-President and Secretary remain the same: - Vice-President: 3 choices - Secretary: 3 choices Total ways = $1 \times 3 \times 3 = 9$ 6. **Part d: Number of ways if voter does not vote for Eman for Vice-President** Exclude Eman from Vice-President choices: - President: 2 choices - Vice-President: $3 - 1 = 2$ choices (Karam, Menna) - Secretary: 3 choices Total ways = $2 \times 2 \times 3 = 12$